As a last remark: note that \(2^{\mathop {\mathrm {rank}}{\mathrm {Jac}\,{(C_f)}(K)}}\)  is \(\ll 1\)  on average (ordering as usual by height) [1]}, [2]}. This statement is why we tried for the stronger bound [3]} in the elliptic curve case, after which we questioned why a similar bound did not exist in the hyperelliptic curve case.
